Hexadecyltrimethylammonium bromide has a long hydrophobic chain and a polar head group.

The molecule does not dissolve well in either aqueous or organic solvents. In an organic
solvent containing a small amount of water the hexadecyltrimethylammonium bromide
traps the aqueous portion in a micelle sphere with the polar heads facing in
and the non-polar tails facing out. The relative amount of pentanol cosurfactant
controls the size of the micelle.

Mixing hexadecyltrimethylammonium
bromide pentanol micelles of CdCl2 with similar micelles containing
Na2S produces
nanoparticle CdS since the aqueous solution serves as a nanoreactor and the
particles cannot grow bigger than the micelle. The pentanol also acts as a
capping agent to stabilize the CdS particles. The formation of CdS nanoparticles
can be detected by spectroscopy since quantum size effects make the visible
absorption spectra different than that of bulk CdS.

The effective mass model suggests
where r is the radius of the nanoparticle. The second term is the particle-in-a-box confinement energy for an electron-hole pair in a spherical quantum dot
and the third term is the Coulomb attraction between an electron and hole modified by the screening of charges by the crystal.

After multiplying by r2, rearranging, and using the quadratic formula,
What is the diameter of the nanoparticles?